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Integers, Pi, and Number Lines.
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magic:

--- Quote from: ledtester on July 05, 2022, 02:22:30 am ---for the record, I did point this out early on:

--- End quote ---
Well,  :-+ then

This is the only thing that really matters. Irrational numbers are filler junk which is postulated to exist solely for the purpose of ensuring that every Cauchy sequence actually has a limit, because somebody found that to be convenient.

All irrational numbers that one actually calculates with are simply solutions to some equations and they could be regarded as such without insisting that they are "numbers". Indeed, when you handle them, you don't process the numbers, you process the equations they are solution to and thus sqrt(3)·sqrt(5) becomes sqrt(15). And if you need a real-world actionable result, you just pick a nearby rational like 3.87298334620741688518.

So OP isn't even far off the mark, but haters gonna hate.


--- Quote from: nigelwright7557 on July 04, 2022, 10:26:36 pm ---I have used Fourier transforms in a scope I designed.
It simply outputs buckets of numbers representing frequency and amplitude.
I didn't need to understand how it worked just what needed to be input and what the outputs meant.
--- End quote ---
Your FFT function only works with rational numbers, too.
ebastler:

--- Quote from: magic on July 05, 2022, 08:27:52 am ---So OP isn't even far off the mark, but haters gonna hate.

--- End quote ---

Hmm, not sure which original post you read, but the one I see finishes with the following. Which I don't hate, but find rather dumb:


--- Quote from: Peter Taylor on June 29, 2022, 11:01:29 pm ---So, a number line should contain only integer numbers.
QED.

--- End quote ---
eugene:

--- Quote from: magic on July 04, 2022, 10:04:15 pm ---Two pages and still no one has pointed out OP's fundamental error:

Almost all real numbers cannot be described as a solution to any equation, and yet mathematicians will insist that they are there >:D

--- End quote ---

I propose that we stop using a base-10 number system and switch to one that is base-pi. Now, when we draw a number line '1' will land exactly where pi used to. Pi does have an exact value. We couldn't write it down before, but now we can. It's as simple as this: 1. Now all the mathematicians can stop worrying about whether or not pi is really "there" or if it's just a question that never gets evaluated. It really is "there."

Of course the old decimal 1 is gone now. It's no longer "there" or describable as a solution to an equation. Now the old decimal 1 is just a question that never gets evaluated. Sucks if you want to buy some eggs. But still, I think I should win a prize in mathematics.
kjpye:
Actually, in a base-pi number system, pi would be represented as 10. Not quite as simple, but convenient.
Peter Taylor:

--- Quote from: IanB on July 05, 2022, 03:01:12 am ---
--- Quote from: Peter Taylor on June 29, 2022, 11:01:29 pm ---In classic education, we are taught that a number line contains an infinite number of irrational numbers between each integer. I will show that it should contain only integers.

dy / dx is never evaluated because dx is infinitely small and zero.

When working with Fourier Transforms for example, the terms are introduced into a formulae as a pair to help solve it, and then disappear at the end without ever being evaluated.

The imaginary number "square root -1" is never evaluated, but introduced into a formulae at the start to help solve it, and disappears at the end.

Pi also is not evaluated, but remains the ratio of the circumference of a circle to its radius.

"Square root 2" likewise is not evaluated, but remains a question: "What value multiplied by itself equals 2 ?".

These are termed irrational "numbers", having no exact value.

But these are not numbers, or values, but questions, or formulae.

If we remove all these non-numbers, or questions, or ratios, or any unevaluated formulae from our number line, we are left only with integer numbers.

So, a number line should contain only integer numbers.

QED.

--- End quote ---

For example, 1/7 is a rational number that is not an integer ...


--- End quote ---

I'm cutting a length of pine to a rational length of 24.6 cm's, but I'm also cutting it to an integer length of 246 mm's.

The rational number 0.142857 ... recurring may repeat infinitely but it is still an integer, 1 / 7.

A "nose zing" equals 7 "whisker zings" and I want to cut my timber to a rational length of  3 / 7 th's of a "nose zing", but I'm also cutting it to an integer length of 3 "whisker zings".

A rational number refers to how an integer is represented in our number system, not to the type of number it is.

In LaLa Land, where they use a base 7 number system, 1 / 7 in our base 10 number system is an integer.

My conjecture states that irrational numbers are not numbers, but questions, and that only integers are numbers.

;)

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